Mid-height deflection of a bar - Euler's Formula

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Discussion Overview

The discussion revolves around a homework problem involving the mid-height deflection of a vertical aluminium bar under axial loading, specifically applying Euler's formula and related equations to determine the deflection before reaching the material's plastic yield stress. The scope includes mathematical reasoning and conceptual clarification regarding the application of formulas in structural mechanics.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about the correctness of their calculations and the applicability of the formulas used, noting that their results do not match the provided multiple-choice options.
  • Another participant suggests that the deflection formula used is inappropriate and recommends using the bending stress formula instead, emphasizing the importance of proper unit notation.
  • A different participant mentions that the section is fully plastic at a deflection of 45 mm and questions whether the problem asks for the deflection at which the extreme fibres first reach yield stress or when the entire section has gone plastic.
  • Another participant challenges the interpretation of the question, asking for clarification on how the previous participant arrived at their conclusion regarding the ambiguity of the question.
  • One participant asserts that aluminum does not have a yield plateau and interprets the question as asking for the deflection when only the extreme fibre reaches the tensile yield strength of 250 MPa.

Areas of Agreement / Disagreement

Participants express differing interpretations of the problem, particularly regarding the definition of yield stress in the context of the question. There is no consensus on the correct approach or interpretation of the problem.

Contextual Notes

Participants highlight potential ambiguities in the question regarding the conditions under which deflection is measured, as well as the applicability of certain formulas. There are unresolved assumptions about the material behavior of aluminum and the definitions of yield stress in this context.

SaRaH...
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Homework Statement


I think I've got part of this question but it's multiple choice and nothing that I've got matches any of the options we were given. I'd really appreciate it if you could help me out.

A straight, vertical aluminium bar, 1.0-m in length and 12.5-mm x 4.8-mm in cross
section, is axially loaded until it buckles. Assuming Euler’s formula applies, determine
the mid-height deflection, in millimetres, of the vertical bar before the material attains
its plastic yield stress of 250-MPa.

Homework Equations



P = EIpi2/L2

I = bd3/12

deflection = PL/AE

The Attempt at a Solution



I = (12.5)(4.8)3/12 = 115.2mm4

P = (70*109)(115.2*10-12)*pi2/(1)2 = 79.6N

Then when I tried to get deflection it came out as a huge answer. I'm not certain if that's the right formula I'm using but it's the only one we've used in class so I don't know what else it could be.

The answers we were given were 57mm, 150mm, 31mm, 145mm, 378mm

Sarah
 
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SaRaH...: Hint 1: Your third relevant equation is inapplicable; you instead need the bending stress formula. Hint 2: Bending moment is force times distance. You correctly computed the force.

By the way, always leave a space between a numeric value and its following unit symbol. E.g., 79.59 N, not 79.59N. And, e.g., 250 MPa, not 250-MPa. See the international standard for writing units[/color] (ISO 31-0[/color]).
 
For what it's worth, I found that the section is FULLY plastic when deflection is 45 mm. So you could check whether extreme fibres first reach yield stress at 31 mm. I think the question is a bit ambiguous, but it's a well intentioned question.
 
pongo38: I am thinking your statement currently appears incorrect, unless I am misinterpreting. Would you be able to show how you obtained your answer? I currently did not find the question ambiguous.
 
Is th question asking for the deflection when the extreme fibres FIRST reach their yield stress, or when the whole section has gone plastic (about 50% more moment required)?
 
pongo38: Aluminum has no yield plateau. I interpreted the question as meaning it is asking for the deflection when only the extreme fibre reaches the tensile yield strength, Sty = 250 MPa.
 
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